Chapter 3 - CHAPTER 3 Second Order Linear Dierential...

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CHAPTER 3 Second Order Linear Differential Equations 3.1 Introduction; Basic Terminology and Results Any second order differential equation can be written as F ( x,y,y ± ,y ±± )=0 This chapter is concerned with special yet very important second order equations, namely linear equations. Recall that a frst order linear differential equation is an equation which can be written in the Form y ± + p ( x ) y = q ( x ) where p and q are continuous Functions on some interval I . A second order, linear differential equation has an analogous Form. DEFINITION 1. A second order linear differential equation is an equation which can be written in the Form y ±± + p ( x ) y ± + q ( x ) y = f ( x ) (1) where p, q , and f are continuous Functions on some interval I . The Functions p and q are called the coefficients oF the equation; the Function f on the right-hand side is called the forcing function or the nonhomogeneous term . The term “Forcing Function” comes From applications oF second-order linear equations; the description “nonhomogeneous” is given below. A second order equation which is not linear is said to be nonlinear . Examples (a) y ±± - 5 y ± +6 y =3cos2 x . Here p ( x )= - 5 ,q ( x )=6 ,f ( x )=3cos2 x are continuous Functions on ( -∞ , ). (b) x 2 y ±± - 2 xy ± +2 y = 0. This equation is linear because it can be written in the Form (1) as y ±± - 2 x y ± + 2 x 2 y =0 where p ( x )=2 /x, q ( x /x 2 ( x ) = 0 are continuous on any interval that does not contain x = 0. ±or example, we could take I =(0 , ). 63
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(c) y ±± + xy 2 y ± - y 3 = e xy is a nonlinear equation; this equation cannot be written in the form (1). ± Remarks on “Linear.” Intuitively, a second order differential equation is linear if y ±± appears in the equation with exponent 1 only, and if either or both of y and y ± appear in the equation, then they do so with exponent 1 only. Also, there are no so-called “cross- product” terms, yy ± ,yy ±± ,y ± y ±± . In this sense, it is easy to see that the equations in (a) and (b) are linear, and the equation in (c) is nonlinear. Set L [ y ]= y ±± + p ( x ) y ± + q ( x ) y . If we view L as an “operator” that transforms a twice differentiable function y = y ( x ) into the continuous function L [ y ( x )] = y ±± ( x )+ p ( x ) y ± ( x q ( x ) y ( x ) , then, for any two twice differentiable functions y 1 ( x ) and y 2 ( x ), L [ y 1 ( x y 2 ( x )] = [ y 1 ( x y 2 ( x )] ±± + p ( x )[ y 1 ( x y 2 ( x )] ± + q ( x )[ y 1 ( x y 2 ( x )] = y ±± 1 ( x y ±± 2 ( x p ( x )[ y ± 1 ( x y ± 2 ( x )] + q ( x )[ y 1 ( x y 2 ( x )] = y ±± 1 ( x p ( x ) y ± 1 ( x q ( x ) y 1 ( x y ±± 2 ( x p ( x ) y ± 2 ( x q ( x y 2 ( x ) = L [ y 1 ( x )] + L [ y 2 ( x )] and, for any constant c , L [ cy ( x )] = [ cy ( x )] ±± + p ( x )[ cy ( x )] ± + q ( x )[ cy ( x )] = cy ±± ( x p ( x )[ cy ± ( x )] + cq ( x ) y ( x ) = c [ y ±± ( x p ( x ) y ± ( x q ( x ) y ( x )] = cL [ y ( x )] . Therefore, as introduced in Section 2.1, L is a linear differential operator. This is the real reason that equation (1) is said to be a linear differential equation. ± The Frst thing we need to know is that an initial-value problem has a solution, and that it is unique.
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This note was uploaded on 02/09/2011 for the course MATH 3321 taught by Professor Morgan during the Fall '08 term at University of Houston.

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Chapter 3 - CHAPTER 3 Second Order Linear Dierential...

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